Lemma (Schur’s lemma).

Proof.

Note that both the kernel, ker⁡f, and the image, im⁡f, are G-submodules of V and
W, respectively. Since V is irreducible, ker⁡f is either
trivial or all of V. In the former case, im⁡f is all of W
— also because W is irreducible — and hence f is invertible. In
the latter case, f is the zero map.
∎

One of the most important consequences of Schur’s lemma is the following.